Elementary equivalence is set-theoretically absolute between any two transitive models of set theory; this is also true for the infinitary logics - e.g., $\mathcal{L}_{\omega_1\omega}$ - at least, assuming the models in question have all the relevant infinitary formulas. In both cases, the easiest way to prove this is to prove that the *satisfaction* of a given sentence in a given structure is absolute (note https://arxiv.org/abs/1312.0670 by Joel David Hamkins and Ruizhi Yang, which justifies my restriction to *transitive* models).

This latter statement is of course false for second-order logic: the statements "the Continuum Hypothesis holds" and "this model is countable" are both expressible by second-order sentences, and their truth values depend on the ambient universe, with the former being being a "switch" and the latter a "button" in the sense of the modal logic of forcing.

However, this alone does not show that second-order elementary equivalence, "$\equiv_{II}$", is not absolute. The simplest way I know to show that $\equiv_{II}$ is not absolute is to consider a pair of models $A, B$ whose cardinalities have different set-theoretic properties; for example, take $A$ and $B$ to be pure sets with cardinalities $\aleph_0$ and $\aleph_1$, respectively. Then the former satisfies "I am countable" while the latter does not, and this is expressible by a second-order sentence, so they are not second-order elementarily equivalent; however, by collapsing $\omega_1$ we make $A$ and $B$ isomorphic.

This leaves open, however, two classes questions about the non-absoluteness of second-order elementary equivalence. I'll mention a couple in each class.

First, how much of this depends on cardinality?

Are there equinumerous $A$, $B$ such that the second-order elementary equivalence of $A$ and $B$ is not absolute between transitive models of set theory containing $A$ and $B$?

And the stronger version:

Are there

countablesuch $A$ and $B$?

Second, what kind of "switching behavior" is possible? The example given above shows that we can "turn $\equiv_{II}$ on," but in that example we cannot turn it off again. An example where we can "turn $\equiv_{II}$ off" is the following: take pure sets $A$ and $B$ of cardinalities $\vert A\vert<\vert B\vert$ which are second-order elementarily equivalent; by the pigeonhole principle, we can in fact find such $A$ and $B$ of cardinality $\le 2^{\aleph_0}{}^+.$ Now consider a forcing extension $V[G]$ in which $\vert A\vert$ is made countable but $\vert B\vert$ is not; in $V[G]$, $A\not\equiv_{II}B$. This raises a couple interesting questions.

First, note that in the above example, we can turn $\equiv_{II}$ back on again by collapsing $\vert B\vert$ to $\omega$. So it makes sense to ask:

Suppose $A\equiv_{II} B$ in $V$. Is it the case that for every generic extension $V[G]$, there is a further generic extension $V[G][H]$ in which $A\equiv_{II} B$?

Second, the only examples I've found so far involve collapsing cardinals. This, of course, runs out when everything reaches $\omega$. So we can ask:

Is there a pair of structures $A, B\in V$ and an $\omega$-sequence of models $V=V_0<V_1<V_2< . . . $ (where "$W<W'$" means "$W'$ is a generic extension of $W$") such that $V_i\models A\equiv_{II}B$ exactly when $i$ is even? In general, what sort of 'alternating behavior' is possible?

This is of course a long list of questions, and I don't expect them all to be answered here. Basically, I'm interested in everything around this issue, so I'll accept any answer which helps me understand the general picture of things.